# Rational Power Functions

One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like $f(x)=x^n$ are called power functions, the rational power functions would be those of the form $f(x)= x^{p/q}$ (where $p/q \in \mathbb{Q}$ ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

$y = -(x-5)^{4/3}+1$
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.